212 ANALYTICAL MECHANICS
the potentials due to a negative charge and a negative pole are negative, while the potentials due to a positive charge and a positive pole are positive.
179. Potential Due to Any Distribution of Mass. — When the field of force or the potential field is due to a number of particles (material, electrical, or magnetic), the potential at a point equals the algebraic sum of the potentials due to the various particles. Thus if mi, m^ ra3, etc., be the masses of the particles and n, r2, r3, etc., their distances from the point considered, then the potential at the point is
/ mi
= —ly
Tl
(VII)
When the field is due to a continuous distribution of mass the last equation may be put in the form of an integral. Thus
7= -7 — - (VII7)
JQ T
180. Intensity of the Field. — The intensity at any point of a potential field, or a field of force, is defined as the force experienced by a unit mass when placed at that point.
Let H denote the intensity at a point. Then, if F is the force experienced by a mass m! when placed at that point, we have, by definition,
-> (VIII)
m }
and #= — •
m
--(-}
m'\ds]
ds \m'